Question

Prove V2 +/2 + 12 + 2 cos 88 = 2 cos 0
​

Answer 1

sin(A + A) = sin A cos A + cos A sin A

But A + A is just 2A, and the two terms on the right-hand side are equal. Therefore:

sin 2A = 2 sin A cos A

The cosine formula is just as easy:

cos(A + A) = cos A cos A − sin A sin A

cos 2A = cos² A − sin² A

Though this is valid, it’s not completely satisfying. It would be nice to have a formula for cos 2A in terms of just a sine or just a cosine. Fortunately, we can use sin² x + cos² x = 1 to eliminate either the sine or the cosine from that formula:

cos 2A = cos² A − sin² A = cos² A − (1 − cos² A) = 2 cos² A − 1

cos 2A = cos² A − sin² A = (1 − sin² A) − sin² A = 1 − 2 sin² A

On different occasions you’ll have occasion to use all three forms of the formula for cos 2A. Don’t worry too much about where the minus signs and 1s go; just remember that you can always transform any of them into the others by using good old sin² x + cos² x = 1.

(59) sin 2A = 2 sin A cos A

cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A

hope this answer helpful u